When \(k=1\)  and \(d\ge 3\) , the inequality (REF ) includes the Schrödinger equation \((-\Delta +V)u=0\)  as a special case, of which UCP has been extensively studied over the past several decades by numerous authors. We refer the reader to the survey articles [1]}, [2]}, [3]} and [4]}, and the references therein. For \(V\in L_{\mathrm {loc}}^{d/2}(\Omega )\) , Kenig, Ruiz, and Sogge [5]} proved that the inequality
\(|\Delta u | \le |Vu| \ \ \text{in} \ \ \Omega \) 
